Solve the equation. $\dfrac{dy}{dx}=\dfrac{\sin(x)}{e^y}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\ln(-\cos(x))+C$ (Choice B) B $y=\ln(-\cos(x)+C)$ (Choice C) C $y=\dfrac{\cos(x)-\sin(x)}{e^y}+C$ (Choice D) D $y=\dfrac{\cos(x)-\sin(x)+C}{e^y}$
Explanation: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{\sin(x)}{e^y} \\\\ e^y\,dy&=\sin(x)\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} e^y\,dy&=\sin(x)\,dx \\\\ \int e^y\,dy&=\int \sin(x)\,dx \\\\ e^y&=-\cos(x)+C \\\\ \ln(e^y)&=\ln(-\cos(x)+C) \\\\ y&=\ln(-\cos(x)+C) \end{aligned}$ Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\ln(-\cos(x)+C)$